Data Analysis & Forecasting Faculty of Development Economics

Phung Thanh Binh (2010) 1

NONSTATIONARY & UNIT ROOTS DICKEY-FULLER TESTS

1. DF Test Equations

The simple Dickey-Fuller test is based on the following AR(1) model:

Y t = ρY t-1 + ut (*)

H0: ρ = 1 (Yt has a unit root ~ Yt is nonstationary)

H1: ρ < 1 (Yt does not have a unit root ~ Yt is stationary)

If we subtract Yt-1 from both sides of (*):

Yt – Yt-1 = ρY t-1 – Yt-1 + ut

∆Y t = (ρ - 1)Yt-1 + ut

∆Y t = δY t-1 + ut (1)

H0: ρ = 1 (Yt has a unit root ~ Yt is nonstationary)

H1: ρ < 1 (Yt does not have a unit root ~ Yt is stationary)

Dickey and Fuller (1979) also proposed two alternative regression equations that can be used for testing for the presence of a unit root.

� The first contains a constant in the random walk process as in the following equation:

∆Y t = α + δY t-1 + ut (2)

� The second case is also allow, a non-stochastic time trend in the model, so as to have:

∆Y t = α + γT + δY t-1 + ut (3)

This test does not have a conventional ‘t’ distribution and so we must use special critical values which were originally calculated by Dickey and Fuller.

MacKinnon (1991,1996) tabulated appropriate critical values for each of the three above models and these are presented in Table 1.

� Table 1: Critical values for DF test

Model 1% 5% 10% ∆Y t = δY t-1 + ut -2.56 -1.94 -1.62 ∆Y t = α + δY t-1 + ut -3.43 -2.86 -2.57 ∆Y t = α + γT + δY t-1 + ut -3.96 -3.41 -3.13 Standard critical values -2.33 -1.65 -1.28

Source: Asteriou (2007)

Data Analysis & Forecasting Faculty of Development Economics

Phung Thanh Binh (2010) 2

If the DF statistical value is smaller in absolute terms than the critical value then we reject the null hypothesis of a unit root and conclude that Yt is a stationary process.

2. Performing DF Test in Eviews

Step 1 Open the file DF.wf1 by clicking File/Open/Workfile and then

choosing the file name from the appropriate path

Step 2 Let’s assume that we want to examine whether the series named GDP

contains a unit root. Double click on the series named ‘GDP’ to open

the series window and choose View/Unit Root Test …. In the unit-

root test dialog box that appears, choose the type test (i.e., the

Augmented Dickey-Fuller test) by clicking on it.

Step 3 We then specify whether we want to test for a unit root in the level,

first difference, or second difference of the series. We can use this

option to determine the number of unit roots in the series.

Step 4 We also have to specify which model of the three DF models we wish

to use. For the model given by equation (1) click on ‘none’ in the

dialog box; for the model given by equation (2) click on ‘intercept’;

and for the model given by equation (3) click on ‘intercept and

trend’.

Step 5 Type ‘0’ on the ‘user specified’ below the “lag length’ dialog box.

Step 6 Having specified these options, click <OK> to carry out the test.

Step 7 We reject the null hypothesis of a unit root against the one-sided

alternative if the DF statistic is less than (lies to the left of) the critical

value, and we conclude that the series is stationary.

Source: Asteriou (2007)